Mathematics · 8 modules · 25 steps
Mathematics for the Curious
Mathematics for an adult who wants to understand why it works, not to pass an exam. It starts with what mathematicians think they are doing, then goes to where the numerals actually came from, which is India and the Islamic world rather than Greece, and follows proof, algebra, geometry, calculus, chance and infinity from there. Emmy Noether sits in the algebra module because she is algebra, Ramanujan sits with infinite series, and Ada Lovelace sits with computation. Almost no symbol manipulation is required. The promise is that mathematics will start to look like a human argument you can follow.
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What Mathematics Is
Three attempts to say what the subject actually is, by a working mathematician defending his life, a populariser who can explain anything, and a Fields medallist being precise about abstraction.
A Mathematician's Apology
G.H. HardyHardy at the end of his career on why mathematics is worth doing, and on beauty as the real criterion. Ninety pages, melancholy, and the best statement of the case.
Open the core textA physicist reading and responding to Hardy's argument about mathematical beauty and usefulness.
Open companionThe great six-hundred-page survey. Not a first read, but the reference to keep once the pathway has given you the questions.
The Joy of x
Steven StrogatzStrogatz walks from counting to calculus in short chapters written for people who gave up on the subject. The best on-ramp available.
Strogatz on how he explains mathematics to people who think they hate it.
Open companionMathematics: A Very Short Introduction
Timothy GowersA Fields medallist on what abstraction buys you and why mathematicians care about proof at all. Short, and more demanding than Strogatz in a useful way.
Abstraction and pattern presented as a way of seeing rather than as a technique.
Open companionWhere the Numbers Came From
The numerals you use are Indian, the word algebra is Arabic, and the standard story that runs Greece to Europe skips a thousand years of the actual line of descent.
The Crest of the Peacock
George Gheverghese JosephJoseph's survey of the non-European roots of mathematics: Egypt, Mesopotamia, China, India, the Islamic world and the Maya, argued against the standard Eurocentric narrative.
A panel on the transmission of Indian mathematics westward and what got attributed to whom.
Open companionMathematics in India
Kim PlofkerThe scholarly account of Indian mathematics from the Vedic period to the eighteenth century, including the Kerala school's infinite series two centuries before Newton.
A university lecture on the Indian tradition, its methods and its chronology.
Open companionScience and Islam: A History
Ehsan MasoodCovers al-Khwarizmi and the birth of algebra alongside the astronomy, optics and medicine of the same tradition. The book to read before deciding where algebra came from.
A documentary on al-Khwarizmi's book and how the word algebra got into every language.
Open companionNumber and Proof
The thing that makes mathematics different from every other subject is proof. One book teaches you to build one, one shows you the most famous chase in the subject's history, and one takes you through the great proofs themselves.
How to Prove It
Daniel VellemanThe book that teaches the actual skill: quantifiers, logic, induction, and how a proof is constructed rather than admired. Work the exercises or do not bother.
Open the core textWhat a proof is and the standard techniques, presented by a mathematician at a whiteboard.
Open companionFermat's Last Theorem
Simon SinghThree hundred and fifty years of failure and one man working in an attic for seven years. The best popular book ever written about mathematical research.
The problem stated plainly and why it was so much harder than it looks.
Open companionJourney through Genius
William DunhamTwelve great theorems with their actual proofs, from Hippocrates to Cantor, each set in the situation that produced it. Read one chapter at a time.
How a proof gets built out of connections between fields, which is Dunham's subject throughout.
Open companionAlgebra
Algebra is not the manipulation of letters. It is the study of structure, and the person who understood that most deeply is the one usually left out of the story.
Algebra
Israel Gelfand and Alexander ShenA Russian mathematician's course for schoolchildren, and far better than anything written for adults. Terse, beautiful and workable at the kitchen table.
What structure means in algebra, built up from symmetry to something genuinely strange.
Open companionMathematics for the Nonmathematician
Morris KlineKline teaches the mathematics through the history and the applications, which is the right order for anyone who was told to memorise it first.
The whole subject laid out as a single map, useful for seeing where any given topic sits.
Open companionEmmy Noether's Wonderful Theorem
Dwight E. NeuenschwanderNoether founded modern abstract algebra and proved that every symmetry in physics gives you a conservation law. She was refused a paid post for years because she was a woman.
Noether's theorem explained through what it does to energy conservation, which is the clearest way in.
Open companionGeometry and Shape
The oldest axiomatic system, the discovery that it was not the only one, and the best fictional demonstration of what an extra dimension would feel like.
Elements
EuclidRead the definitions, the postulates and the first book. Two thousand years of mathematicians learned what an argument was from this, and it is shorter than you expect.
Why the fifth postulate bothered people for two millennia, and what happened when it was dropped.
Open companionEuclid's Window
Leonard MlodinowGeometry from Euclid to Einstein to string theory, told as a story about five revolutions in how space was understood.
Curved space made intuitive, which is what Mlodinow's later chapters are asking you to picture.
Open companionFlatland
Edwin A. AbbottA square living in two dimensions meets a sphere and cannot cope. Written in 1884, still the best thing ever written about dimension, and a satire on Victorian society as well.
The same thought experiment run forwards, from a line to a plane to a solid to a fourth dimension.
Open companionChange and Calculus
The mathematics of things that move. One book from 1910 that still teaches it best, one that explains why it matters, and one man who arrived at infinite series from nowhere.
Calculus Made Easy
Silvanus P. ThompsonOpens with the line that what one fool can do, another can. Thompson explains the derivative and the integral without the apparatus, and it works.
The derivative built up visually from area and slope, which is the intuition Thompson is after.
Open companionInfinite Powers
Steven StrogatzWhy calculus is the language the universe turned out to be written in, from Archimedes to HIV drug regimens. No prior calculus needed.
Open the core textStrogatz on why calculus frightens people and what it is actually for.
Open companionThe Man Who Knew Infinity
Robert KanigelRamanujan, a clerk in Madras with almost no formal training, sending Hardy results nobody had seen. The infinite series in his notebooks connect back to the Kerala school.
The documentary on Ramanujan and Hardy, filmed in both Cambridge and Madras.
Open companionChance and Statistics
The part of mathematics you are most likely to be lied to with. Three books that between them cover the reasoning, the practice and where the methods came from.
How Not to Be Wrong
Jordan EllenbergEllenberg on mathematical thinking applied to lottery tickets, opinion polls, obesity trends and the Baltimore stockbroker. Funny, and it changes how you read the news.
Ellenberg in conversation about the kind of reasoning the book is trying to teach.
Open companionThe Art of Statistics
David SpiegelhalterHow to learn from data honestly, by the statistician who spends his time correcting newspapers. Almost no formulae and a great deal of judgement.
Open the core textSpiegelhalter on communicating risk and on the ways numbers get used to mislead.
Open companionThe Lady Tasting Tea
David SalsburgHow twentieth-century statistics was actually invented, starting with Fisher designing an experiment to test whether a woman could tell how her tea was poured.
The founding experiment reconstructed, which shows what a significance test is really doing.
Open companionInfinity, Logic and Computation
Where mathematics turns on itself and asks what it can and cannot do. Four books on the sizes of infinity, the limits of proof, the first program, and the problem nobody has solved.
Everything and More
David Foster WallaceA novelist on Cantor and the infinite, written with obsessive footnotes and real mathematical content. Uneven, and nothing else reads like it.
A public lecture on how Cantor proved that some infinities are larger than others.
Open companionGodel's Proof
Ernest Nagel and James R. NewmanA hundred pages on the incompleteness theorems, which showed that any system strong enough to do arithmetic contains truths it cannot prove.
Godel, Turing and the halting problem in one narrative, with the diagonal argument drawn out.
Open companionAda's Algorithm
James EssingerAda Lovelace saw that Babbage's engine could operate on things other than numbers, which is the idea the whole of computing rests on. She wrote it down in 1843.
What the machine was and what Lovelace understood about it that Babbage did not.
Open companionThe Music of the Primes
Marcus du SautoyThe Riemann hypothesis, the primes, and a century and a half of mathematicians failing to prove the most important open problem in the subject.
The hypothesis stated so you can see what it claims and why the primes are involved.
Open companion